Faithfulness conjecture for the Kontsevich integral of bottom tangles in handlebodies

Let $\bfF$ denote the category of finitely generated free groups. Consider the functor

$$h:\B\cong\mathcal{H}^{\operatorname{op}}\lto \bfF^{\operatorname{op}}$$

that assigns to each bottom tangle $T:m\to n$ the homomorphism $(i_T)_*:F_n\to F_m$. The functor $Z^\B$ is the Kontsevich integral for bottom tangles in handlebodies, and $\Zq^\vp$ is its restriction or completion on the corresponding linearized category. Faithfulness conjecture. The functor $Z^\B$ (respectively, $\Zq^\vp$) is faithful; equivalently, it is a complete invariant of bottom tangles in handlebodies. The conjecture is motivated by the expected ability of universal Vassiliev–Goussarov invariants to distinguish knots, but the paper does not establish faithfulness.